We investigate the formal deformation theory of (rank 1) branes on generalized complex (GC) manifolds. This generalizes, for example, the deformation theory of a complex submanifold in a fixed complex manifold. For each GC brane $\mathcal{ B}$ on a GC manifold $(X,\mathbb{ J})$, we construct a formal (pointed) groupoid $\textbf{ Def}^{ \mathcal{ B}}(X,\mathbb{ J})$ (defined over a certain category of real Artin algebras) that encodes the formal deformations of $\mathcal{ B}$. We study the geometric content of this groupoid in a number of different situations. Using the theory of (bi)semicosimplicial differential graded Lie algebras (DGLAs), we construct for each brane $\mathcal{ B}$ a DGLA $L_{ \mathcal{ B}}$ that governs the "locally trivializable" deformations of $\mathcal{ B}$. As a concrete application of this construction, we prove an unobstructedness result.
Categories: math.DG math.AG